nLab D=4 N=2 super Yang-Mills theory

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Context

Physics

physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Quantum field theory

Contents

Idea

The special case of super Yang-Mills theory over a spacetime of dimension 4 and with N=2N = 2 supersymmetry.

ddNNsuperconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
via AdS-CFT
A3A\phantom{A}3\phantom{A}A2k+1A\phantom{A}2k+1\phantom{A}AB(k,2)\phantom{A}B(k,2) \simeq osp(2k+1|4)A(2k+1 \vert 4)\phantom{A}ASO(2k+1)A\phantom{A}SO(2k+1)\phantom{A}
A3A\phantom{A}3\phantom{A}A2kA\phantom{A}2k\phantom{A}AD(k,2)\phantom{A}D(k,2)\simeq osp(2k|4)A(2k \vert 4)\phantom{A}ASO(2k)A\phantom{A}SO(2k)\phantom{A}M2-brane
D=3 SYM
BLG model
ABJM model
A4A\phantom{A}4\phantom{A}Ak+1A\phantom{A}k+1\phantom{A}AA(3,k)𝔰𝔩(4|k+1)A\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
A5A\phantom{A}5\phantom{A}A1A\phantom{A}1\phantom{A}AF(4)A\phantom{A}F(4)\phantom{A}ASO(3)A\phantom{A}SO(3)\phantom{A}D4-brane
D=5 SYM
A6A\phantom{A}6\phantom{A}AkA\phantom{A}k\phantom{A}AD(4,k)\phantom{A}D(4,k) \simeq osp(8|2k)A(8 \vert 2k)\phantom{A}ASp(k)A\phantom{A}Sp(k)\phantom{A}M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

Properties

Moduli space of vacua

A speciality of N=2N=2, D=4D = 4 SYM is that its moduli space of vacua has two “branches” called the Coulomb branch and the Higgs branch. This is the content of what is now called Seiberg-Witten theory (Seiberg-Witten 94). Review includes (Albertsson 03, section 2.3.4).

Confinement

While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), N=2 D=4 super Yang-Mills theory is has been obsrved in (Seiberg-Witten 94).

Reduction to D=3D = 3 super Yang-Mills

By dimensional reduction on 3×S 1\mathbb{R}^3 \times S^1 families of N=2,D=4N = 2, D = 4 SYM theories interpolate to N=4 D=3 super Yang-Mills theory. (Seiberg-Witten 96).

Construction by compactification of 5-branes

N=2N=2 super Yang-Mills theory can be realized as the worldvolume theory of M5-branes compactified on a Riemann surface (Klemm-Lerche-Mayr-Vafa-Warner 96, Witten 97, Gaiotto 09), hence as a compactifiction of the 6d (2,0)-superconformal QFT on the M5. This in particular gives a geometric interpretation of Seiberg-Witten duality in 4d in terms of the 6d 5-brane geometry.

Specifically the AGT correspondence expresses this relation in terms of the partition function of the theory and a 2d CFT on the Riemann surface on which the 5-brane is compactified. See at AGT correspondence for more on this.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Loc GLoc_G, geometric Langlands correspondence
ddNNsuperconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
via AdS-CFT
A3A\phantom{A}3\phantom{A}A2k+1A\phantom{A}2k+1\phantom{A}AB(k,2)\phantom{A}B(k,2) \simeq osp(2k+1|4)A(2k+1 \vert 4)\phantom{A}ASO(2k+1)A\phantom{A}SO(2k+1)\phantom{A}
A3A\phantom{A}3\phantom{A}A2kA\phantom{A}2k\phantom{A}AD(k,2)\phantom{A}D(k,2)\simeq osp(2k|4)A(2k \vert 4)\phantom{A}ASO(2k)A\phantom{A}SO(2k)\phantom{A}M2-brane
D=3 SYM
BLG model
ABJM model
A4A\phantom{A}4\phantom{A}Ak+1A\phantom{A}k+1\phantom{A}AA(3,k)𝔰𝔩(4|k+1)A\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
A5A\phantom{A}5\phantom{A}A1A\phantom{A}1\phantom{A}AF(4)A\phantom{A}F(4)\phantom{A}ASO(3)A\phantom{A}SO(3)\phantom{A}D4-brane
D=5 SYM
A6A\phantom{A}6\phantom{A}AkA\phantom{A}k\phantom{A}AD(4,k)\phantom{A}D(4,k) \simeq osp(8|2k)A(8 \vert 2k)\phantom{A}ASp(k)A\phantom{A}Sp(k)\phantom{A}M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

References

Introductions and surveys

General

The terminology “Coulomb branch” and “Higgs branch” first appears in

See also at Seiberg-Witten theory:

The dimensional reduction to D=3D = 3 was first considered in

The confinement-phenomenon was observed in

Reviews of this confinement mechanism:

For references on wall crossing of BPS states see the references given there.

Construction from 5-branes

N=2N=2 D=4D=4 SYM including its Seiberg-Witten theory (Seiberg-Witten 94) may be understood as being the compactification of the 6d (2,0)-superconformal QFT on the worldvolume of M5-branes on a Riemann surface: the Riemann surface is identified with the Seiberg-Witten curve of complexified coupling constants. This observation goes back to

review in:

The further observation that therefore the sewing of Riemann surfaces on which one compactifies the M5-brane yields a gluing operation on N=2 SYM theories is due to

cf. at AGT correspondence.

The topological twisting of the compactification which is used around (2.27) there was previously introduced in section 3.1.2 of:

and is discussed also for instance in section 5.1 of

(This is possibly also the mechanism behind the AGT correspondence, though the details behind that statement seem to be unclear.)

A brief review of these matters is in (Moore 12, section 7). A formalization of the topological twist in perturbation theory formalized by factorization algebras with values in BV complexes is in section 16 of

For more on this see at topologically twisted D=4 super Yang-Mills theory.

An amplification of the relevance of this to the understanding of S-duality/electric-magnetic duality is in

and the resulting relation to the geometric Langlands correspondence is discussed in

The corresponding dual theory under AdS-CFT duality is discussed in

Discussion of construction of just N=1 D=4 super Yang-Mills theory this way is in

Relation to 3-branes in M5-branes (M5 self-intersections)

Relation to the 3-brane in 6d:

and via F-theory in

  • Robert de Mello Koch, Alastair Paulin-Campbell, Joao P. Rodrigues, Monopole Dynamics in 𝒩=2\mathcal{N}=2 super Yang-Mills Theory From a Threebrane Probe, Nucl. Phys. B559 (1999) 143-164 (arXiv:hep-th/9903207)

Last revised on January 9, 2024 at 09:22:12. See the history of this page for a list of all contributions to it.